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Lévy processes and Itô-Skorohod integrals
Khalifa Es-Sebaiy, Ciprian Tudor
To cite this version:
Khalifa Es-Sebaiy, Ciprian Tudor. Lévy processes and Itô-Skorohod integrals. Theory of Stochastic
Processes, John Wiley & Sons, 2008, 14 (30) (2), pp.10-18. �hal-00308764�
L´EVY PROCESSES AND IT ˆO-SKOROHOD INTEGRALS
Khalifa Es-Sebaiy and Ciprian A. Tudor
Abstract. We study Skorohod integral processes on L´evy spaces and we prove an equivalence between this class of processes and the class of Itˆo-Skorohod processes (in the sense of [14]). Using this equivalence we introduce a stochastic analysis of Itˆo type for anticipating integrals on L´evy spaces.
Introduction
We study in this work anticipating integrals with respect to a L´evy process. The anticipating integral on the Wiener space, known in general as the Skorohod integral (and sometimes as the Hitsuda integral) constitutes an extension of the standard Itˆo integral to non-adapted integrands. It is nothing else than the classical Itˆo integral if the integrand is adapted. The Skorohod integral has been extended to the Poisson process and next it has been defined with respect to a normal martingale (see [3]) due to the Fock space structure generated by such processes. Recently, an anticipating calculus of Malliavin-type has been defined on L´evy spaces again by using some multiple stochastic integrals with respect to a L´evy process which have been in essence defined in the old paper by K. Itˆo (see [4]). We refer to [8], [9] or [13] for Malliavin calculus on L´evy spaces and possible applications to mathematical finance.
The purpose of this paper is to understand the relation between anticipating Skorohod integral pro-cesses and Itˆo-Skorohod integral propro-cesses (in the sense of [14] or [11]) in the L´evy case. We recall that the results in [14] and [11] show that on Wiener and Poisson spaces the class of Skorohod integral pro-cesses with regular enough integrands coincides with the class of some Itˆo-Skorohod integrals that have similar properties to the classical Itˆo integrals for martingales. The fact that the driven processes have independent increments plays a crucial role. Therefore, it is expected to obtain the same type of results for L´evy processes. We prove here an equivalence between Skorohod and Itˆo-Skorohod integrals by using the recent introduced Malliavin calculus for L´evy processes. Some direct consequences of the equivalence between the two classes of stochastic processes are also obtained.
Section 2 contains some preliminaries on L´evy processes and Malliavin-Skorohod calculus with respect to them. In Section 3 we prove a generalized Ocone-Clark formula that we will use in Section 4 to prove the correspondence between Skorohod and Itˆo-Skorohod integrals and to develop an Itˆo-type calculus for anticipating integrals on L´evy spaces.
Preliminaries
In this section we introduce the basic properties of Malliavin calculus for L´evy processes that we will need in the paper. For more details the reader is referred to [13].
In this work we deal with a c`adl`ag L´evy process X = (Xt)0≤t≤1defined on certain complete probability
space ¡Ω, (FX
t )0≤t≤1, P¢, with the time horizon T = [0, 1], and equipped with its generating triplet
1991 Mathematics Subject Classification. 60H07, 60G20, 60G48, 60J75..
Key words and phrases. Malliavin Calculus, martingale-valued measure, Skorohod integral, Clark-Ocone formula.. Typeset by AMS-TEX
(γ, σ2, ν) where γ ∈ R, σ ≥ 0 and ν(dz) is the L´evy measure on R which, we recall, is such that
ν({0}) = 0 and Z
R
1 ∧ x2ν(dx) < ∞
Throughout the paper, we suppose that RRx2ν(dx) < ∞, and we use notation and terminologies as in [1], [13]. By N we will denote the jump measure of X:
N (E) = #{t : (t, ∆Xt) ∈ E},
for E ∈ B(T × R0) , where R0= R − {0}, ∆Xt= Xt− Xt−, # denotes the cardinal. We will note eN the
compensated jump measure:
e
N (dt, dx) = N (dt, dx) − dtν(dx). The process X admits a L´evy-Itˆo representation
Xt= γt + σWt+ Z Z (0,t]×{|x|>1} xN (ds, dx) + lim ε↓0 Z Z (0,t]×{ε<|x|≤1} x eN (ds, dx)
where W is a standard Brownian motion.
Itˆo [4] proved that X can be extended to a martingale-valued measure M of type (2, µ) on (T × R, B(T × R)):
For any E ∈ B(T × R) with µ(E) < ∞
M (E) = σ Z E(0) dWs+ lim n→∞ Z Z {(s,x)∈E:1 n<|x|<n} x eN (ds, dx).
where E(0) = {s ∈ T : (s, 0) ∈ E} and
µ(E) = σ2 Z E(0) ds + Z Z {E−E(0)×{0}} x2dsν(dx).
Furthermore, M is a centered independent random measure such that E (M (E1)M (E2)) = µ(E1∩ E2)
for any E1, E2∈ B(T × R) with µ(E1) < ∞ and µ(E2) < ∞.
Using the random measure M one can construct multiple stochastic integrals driven by a L´evy process as an isometry between L2(Ω) and the space L2((T × R)n, B((T × R)n), µ⊗n). Indeed, one can use the
same steps as on the Wiener space: first, consider a simple function f of the form f = 1E1×...×En
where E1, . . . , En ∈ B(T × R) are pathwise disjoint and µ(Ei) < ∞ for every i. For a such function
define
In(f ) = M (E1) . . . M (En))
and then the operator In can be extended by linearity and continuity to an isometry between L2(Ω) and
L´EVY PROCESSES AND SKOROHOD INTEGRALS 3
An interesting fact is that, as in the Brownian and Poissonian cases, M enjoys the chaotic representa-tion property (see [13]), i.e. for every F ∈ L2(Ω, FX, P ) = L2(Ω), can be written as an orthogonal sum
of multiple stochastic integrals
F = E(F ) +
∞
X
n=1
In(fn)
where this converges in L2(Ω) and f
n ∈ L2s((T × R)n, B((T × R)n), µ⊗n) (the last space is the space of
symmetric and square integrable functions on (T × R)n with respect to µ⊗n.)
At this point we can introduce a Malliavin calculus with respect to the L´evy process X by using this Fock space-type structure. If
∞
X
n=0
nn!kfnk2n < ∞
(here kfnkndenotes the norm in the space L2((T × R)n, B((T × R)n), µ⊗n)) then the Malliavin derivative
of F is introduced as an annihilation operator (see for example [7])
DzF = ∞
X
n=1
nIn−1(fn(z, .)), z ∈ T × R.
The domain of derivative operator D is denoted by D1,2. It contains the random variable of the above
chaotic form such that P∞n=0nn!kfnk2n < ∞ holds. We denote by Dk,2, k ≥ 1 the domain of the kth
iterated derivative D(k), which is a Hilbert space with respect the scalar product
hF, Gi = E(F G) + k X j=1 E Z (T ×R)j D(j) z F Dz(j)Gµ⊗j(dz).
We introduce now the Skorohod integral with respect to X as a creation operator. Let u ∈ H = L2¡T × R × Ω, B(T × R) ⊗ FX
T , µ ⊗ P
¢
, then, for every z ∈ T × R, u(z) admit the following representa-tion u(z) = ∞ X n=0 In(fn(z, .)). Here we have fn∈ L2((T × R)n+1, µ⊗ n+1
) and fn is symmetric in the last n variables. If ∞
X
n=0
(n + 1)!k ˜fnk2n+1< ∞
( ˜fn represents the symmetrization of fn in all its n + 1 variables) then the Skorohod integral δ(u) of u
with respect to X is introduced by
δ(u) =
∞
X
n=0
In+1( ˜fn).
The domain of δ is the set of processes satisfying P∞n=0(n + 1)!k ˜fnk2n+1 < ∞ and we have the duality
relationship
E(F δ(u)) = E Z Z
T ×R
We will use the notation δ(u) = Z 1 0 Z R uzδM (dz) = Z 1 0 Z R us,xδM (ds, dx).
Remark 1. It has been proved in [13] that if the integrand is predictable then the Skorohod integral coincides with the standard semi-martingale integral introduced in [1].
For k ≥ 1, we denote by Lk,2 the set L2((T × R; Dk,2), µ). In particular one can prove that L1,2 is
given by the set of u in the above chaotic form such that
∞
X
n=0
(n + 1)!k ˜fnk2n+1< ∞.
We also have Lk,2⊂ Domδ for k ≥ 1 and for every u, v ∈ L1,2
E(δ(u)δ(v)) = E Z Z T ×R u(z)v(z)µ(dz) + E Z Z (T ×R)2 Dzu(z′)Dz′v(z)µ(dz)µ(dz′). In particular E(δ(u))2= E Z Z T ×R u(z)2µ(dz) + E Z Z (T ×R)2 Dzu(z′)Dz′u(z)µ(dz)µ(dz′).
The commutativity relationship between the derivative operator and Skorohod integral is given by: let u ∈ L1,2 such that D
zu ∈ Dom(δ), then δ(u) ∈ D1,2 and
Dzδ(u) = u(z) + δ(Dz(u)), z ∈ T × R.
Generalized Clark-Ocone formula
We start this section by proving some properties of the multiple integrals In(f ) and how it behaves
if it is conditioned by a σ-algebra. If A ∈ B(T ) we will denote by FX
A the σ -algebra generated by the
increments of the process X on the set A
FAX = σ(Xt− Xs: s, t ∈ A).
Proposition 1. The text of the Proposition Let fn ∈ L2s((T × R)n, µ⊗n) and A ∈ B(T ). Then
E¡In(f )/FAX
¢
= In(f 1⊗n(A×R)).
Proof. By density and linearity argument, it is enough to consider f = 1E1×...×En, where E1, ..., En are pairwise disjoint set of B (T × R) and µ(Ei) < ∞ for every i = 1, ..., n. In this case we have
E¡In(f )/FAX ¢ = E¡M (E1)....M (En)/FAX ¢ E ÃYn i=1 (M (Ei∩ (A × R)) + M (Ei∩ (Ac× R))) /FAX ! = n Y i=1 M (Ei∩ (A × R)) = In(f 1⊗ n (A×R)).
L´EVY PROCESSES AND SKOROHOD INTEGRALS 5
Corollary 1. Suppose that F ∈ D1,2 and A ∈ B(T ). Then the conditional expectation E(F/FX
A) belongs
to D1,2 and for every z ∈ T × R we have that
DzE¡F/FAX
¢
= E¡DzF/FAX
¢
1A×R(z).
Proof. Let F =Pn≥0In(fn) with fn∈ L2s((T × R)n, B((T × R)n), µ⊗n). Then by Proposition 1,
DzE ¡ F/FAX ¢ = Dz X n≥0 In ¡ fn1⊗nA×R ¢ =X n≥1 nIn−1 ³ fn(·, z)1⊗(n−1)A×R ´ 1A×R(z)
and it remains to observe that E¡DzF/FAX
¢ =Pn≥1nIn−1 ³ fn(·, z)1⊗(n−1)A×R ´ .
At this point we will state the following version of Ocone-Clark formula on L´evy space which extends a results in [13].
Proposition 2. [Generalized Clark-Ocone-Haussman formula] Let F be a random variable in D1,2. Then
for every 0 ≤ s < t ≤ 1, we have
F = E³F/F(s,t]X c ´
+ δ(hs,t(·))
where for (r, x) ∈ T × R we denoted by hs,t(r, x) = E¡Dr,xF/F(r,t]c¢1(s,t]c(r). Moreover
F = E³F/F(s,t]X c ´ + Z Z (s,t]×R (p,t)(D zF ) dMz = E³F/FX (s,t]c ´ + σ Z t s (p,t)(D r,0F )dWr+ Z Z (s,t]×R0 (p,t)(D r,xF ) eN (dr, dx)
where(p,t)(DF ) is the predictable projection of DF with respect to the filtration³FX (r,t]c
´
r≤t.
Proof. Let F =P∞n=0In(fn), where fn ∈ L2s
¡
([0, 1] × R)n, µ⊗n¢
. Firstly, we prove that
F = E³F/F(s,t]X c ´
+ δ (hs,t(·)) .
Indeed, for any s < t ≤ 1 we have
E(Dr,xF/F(r,t]X c)1(s,t]×R(r, x) = ∞ X n=1 nIn−1 h fn((r, x), .)1⊗ n−1 (r,t]c×R(.)1(s,t]×R(r, x) i
Hence, using that x ∈ R and thus the symmetrization with respect to the variable x has no effect, we obtain δ (h) = ∞ X n=1 nIn · fn((t1, x1), ..., (tn, xn))1⊗ ^ n−1 (t1,t]c(t2, ..., tn)1(s,t](t1) ¸
Since ^ 1⊗(tn−1 1,t]c(t2, ..., tn)1(s,t](t1) = 1 n! n X i=1 n X σ(1)=i,σ∈Sn 1⊗(tn−1 i,t]c(tσ(2), ..., tσ(n))1(s,t](ti) = 1 n n X i=1 1⊗(tn−1 i,t]c(t1, ..., bti, ..., tn)1(s,t](ti) = 1 n ³ 1 − 1⊗(s,t]nc(t1, ..., tn) ´ . Then δ (hs,t) = F − E ³ F/FX (s,t]c ´ .
The second equality in the statement follows from [13] (see also [11]) where the equivalence of the two representations has been proven.
Itˆo-Skorohod integral calculus
As a consequence of the above results, we will show in this part that every Skorohod integral process of the form
Yt:= δ(u.1[0,t]×R(·)), t ∈ T
can be written as an Itˆo-Skorohod integral in the sense of [14] (this integral has similarities with the standard stochastic integral). We will extend in this way the results of [14] in the Wiener case and of [11] in the Poisson case. The key point of our construction is the fact that the driving process has independent increments.
Proposition 3. Assume that u ∈ Lk,2 with k ≥ 3, then there exist an unique process v ∈ Lk−2,2 such
that for every t ∈ T
Yt:= δ(u.1[0,t]×R(·)) = Z Z (0,t]×R (p,t)(v s,x) M (ds, dx). Moreover vs,x= Ds,xYs µ ⊗ P a.e. on T × R × Ω.
Proof. Applying above generalized Clark-Ocone formula we have
Yt= E ¡ Yt/FtXc ¢ + Z Z (0,t]×R (p,t)(D zYt) dMz
The process Y satisfies
E³Yt− Ys/F(s,t]X c ´
= 0 for every s < t. Indeed, we take a random variable FX
(s,t]c-measurable F in D1,2. According to the duality
relationship and Corollary 1, we have
E(F (Yt− Ys)) = E[δ(u.1(s,t]×R(.))] = E D.F, u.1(s,t]×R(.) ® L2(T ×R, µ)= 0. Therefore, we obtain E¡Yt/FtXc ¢ = E¡Yt− Y0/FtXc ¢ = 0,
L´EVY PROCESSES AND SKOROHOD INTEGRALS 7 and δhE (p,t)(D s,xYt) 1[0,t]×R(s, x) i = δhE³Ds,xYt/F(s,t]X c ´ 1[0,t]×R(s, x) i = δhDs,xE ³ Yt/F(s,t]X c ´ 1[0,t]×R(s, x) i = δhDs,xE ³ Ys/F(s,t]X c ´ 1[0,t]×R(s, x) i = δhE³Ds,xYs/F(s,t]X c ´ 1[0,t]×R(s, x) i = δhE (p,t)(Ds,xYs) 1[0,t]×R(s, x) i We thus have Yt= Z Z (0,t]×R (p,t)(D s,xYs) M (ds, dx).
Set vs,x = Ds,xXs. To obtain the desired Itˆo-Skorohod representation it is sufficient to prove that
v ∈ Lk−2,2. By using the property of commutativity between D and δ and the inequalities for the norms
of anticipating integrals we have
kvk21,2 ≤ kuk21,2+ kδ(Ds,xu.1[0,s]×R(.))k21,2 ≤ kuk21,2+ E Z Z T ×R ¡ δ(Ds,xu.1[0,s]×R(.)¢ 2 µ(ds, dx) +E Z Z (T ×R)2 ¡ Dr,yδ(Ds,xu.1[0,s]×R(.))¢ 2 µ(ds, dx)µ(dr, dy) ≤ kuk21,2+ 3E Z T ×R Z T ×R (Dz2uz1) 2 µ(z1)µ(z2) +3E Z T ×R Z T ×R Z T ×R (Dz3Dz2uz1) 2 µ(z1)µ(z2)µ(z3) +2E Z T ×R Z T ×R Z T ×R Z T ×R (Dz4Dz3Dz2uz1) 2 µ(z1)µ(z2)µ(z3)µ(z4) ≤ 4kuk23,2.
In the same manner, we found that
kvk2k−2,2 ≤ C(k)kuk2k,2,
where C(k) is a positive constant depending of k. To conclude our proof we have to show the uniqueness of these processes. We assume that there exist v and v′ in Lk−2,2 such that
Yt= Z Z (0,t]×R (p,t)(v s,x) M (ds, dx) = Z Z (0,t]×R (p,t)(v′ s,x) M (ds, dx).
Using again the property of commutativity, we have E³ws,x/F[s,t]X c ´ 1[0,t]×R(s, x) + Z Z (s,t]×R E³Ds,xwr,x/F[r,t]X c ´ M (dr, dx) = 0,
where ws,x= vs,x− vs,x′ . Conditioning by F[s,t]X c, we obtain
E³ws,x/F[s,t]X c ´
1[0,t]×R(s, x) = 0, s ≤ t, x ∈ R.
By letting t goes to s we get that ws,x = 0 in L2(Ω) for every (s, x) ∈ T × R. We can thus conclude
that v = v′ in Lk−2,2.
Using the correspondence between Skorohod and Itˆo-Skorohod integrals proved above, we can derive an Itˆo formula for anticipating integrals on L´evy space. As far as we know, this is the only Itˆo formula proved for this class of stochastic processes.
Proposition 4. [Itˆo’s formula] Let v be a process belonging to L2(T × R × Ω, µ ⊗ P ). Let us consider
the following stochastic process
Yt= Z Z (0,t]×R E³vs,x/F[s,t]X c ´ M (ds, dx)
and let f be a C2 real function. Then
f (Yt) = f (0) + Z Z (0,t]×R f′(Ys− t )(p,t)(Ds,xYs) M (ds, dx) +1 2 Z Z (0,t]×R f′′(Yts−)((p,t)(Ds,0Ys))2 ds + X 0<s≤t (f (Yts) − f (Ys − t ) − f′(Ys − t )(Yts− Ys − t )) where Ys t := R R (0,s]×RE ³ vs,x/F[s,t]X c ´ M (ds, dx) and Ys−
t = limr→s−Ytr for all 0 < s ≤ t.
Proof. Fix t ∈ (0, T ]. We define Zs= Yts if s ≤ t and Zs= Yt if s > t. Also let (Gs)s≥0 be a filtration
given as follows Gs= F[s,t]X c if s ≤ t and Gs= F
X
1 , if s > t.
It is easy to see that (Zs)s≥0 is a square integrable cadlag martingale with respect to (Gs)s≥0 and
therefore the standard stochastic calculus for jump processes can be applied to it. Applying Itˆo’s formula (see [12], Theorem 32, p. 71) we obtain for every s > 0
f (Zs) = f (0) + Z (0,t] f′(Z s−)dZs+ 1 2 Z (0,t] f′′(Z s−)d[Z, Z]cs + X 0<s≤t (f (Zs) − f (Zs−) − f′(Zs−)(Zs− Zs−))
L´EVY PROCESSES AND SKOROHOD INTEGRALS 9
where [Z, Z]c is the continuous part of quadratic variation process [Z, Z] of Z. It is well known that
[N, N ]c
s= 0. From Proposition 4 of [13], we see that for every s ≤ t
[Z, Z]c s= [Y, Y ]cs= σ2 Z (0,s] E³vr,0/F[r,t]X c ´2 dr.
Thus, and in particular for s = t(in the sense of the limit almost sure or in L2))
f (Zt) = f (Yt) = f (0) + Z Z (0,t]×R f′(Yts−)E ³ vs,x/F[s,t]X c ´ M (ds, dx) +1 2 Z Z (0,t]×R f′′(Ys− t ) h E³vs,0/F[s,t]X c ´i2 ds + X 0<s≤t (f (Yts) − f (Ys − t ) − f′(Ys − t )(Yts− Ys − t )).
So that the result holds.
Another consequence of Proposition 3 is the following Burkholder inequality which gives a bound for the Lp norm of the anticipating integral.
Proposition 5. [Burkholder’s Inequality] Let Y be a process of Itˆo-Skorohod form as above and 2 ≤ p < ∞. Then there exist a universal constant C(p) such that
E|Yt|p≤ C(p)E Ã σ2 Z (0,t] E³vr,0/F[r,t]X c ´2 dr + Z Z (0,t]×R0 x2E³vr,x/F[r,t]X c ´2 N (dr, dx) !p/2
Proof. The proof of this proposition is straightforward from Theorem 54, P. 174 in [12] and the approxi-mation procedure used along the paper.
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E-mail address: k.essebaiy@ucam.ac.ma
SAMOS/MATISSE, Centre d’Economie de La Sorbonne, Universit´e de Panth´eon-Sorbonne Paris 1, 90, rue de Tolbiac, 75634 Paris Cedex 13, France.
